Hyper-Mahler measures via Goncharov-Deligne cyclotomy

Abstract

The hyper-Mahler measures $m_k(1+x_1+x_2),k\in {\mathbb{Z}}_{>1}$ and $m_k(1+x_1+x_2+x_3),k\in {\mathbb{Z}}_{>1}$ are evaluated in closed form via Goncharov-Deligne periods, namely $\mathbb{Q}$-linear combinations of multiple polylogarithms at cyclotomic points (complex-valued coordinates that are roots of unity). Some infinite series related to these hyper-Mahler measures are also explicitly represented as Goncharov-Deligne periods of levels $1$, $2$, $3$, $4$, $6$, $8$, $10$ and $12$.

arXiv comment(v1) i+30 pages, 5 tables. (v2) i+37 pages, 7 tables. Results improved and enriched. Maple and Mathematica worksheets available as ancillary files. (v3) 47 pages, 8 tables. Reformatted and corrected. (v4) 51 pages, 8 tables. Accepted version

References in corpus (31)

• Harmonic Polylogarithms
• The analytical value of the electron (g-2) at order alpha^3 in QED
• HPL, a Mathematica implementation of the harmonic polylogarithms
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• The Multiple Zeta Value Data Mine
• PolyLogTools - Polylogs for the masses
• Massive Feynman diagrams and inverse binomial sums
• New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams
• The massless higher-loop two-point function
• Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity
• Extension of HPL to complex arguments
• A Feynman integral via higher normal functions
• Single-scale diagrams and multiple binomial sums
• Densities of short uniform random walks
• On the reduction of generalized polylogarithms to ${\text{Li}}_n$ and ${\text{Li}}_{2,2}$ and on the evaluation thereof
• From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop
• On the all-order epsilon-expansion of generalized hypergeometric functions with integer values of parameters
• Some examples of Mahler measures as multiple polylogarithms
• Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes
• Higher Mahler measures and zeta functions
• On a problem involving the squares of odd harmonic numbers
• Multiple zeta values, WZ-pairs and infinite sums computations
• Finite and symmetrized colored multiple zeta values
• On two conjectures of Sun concerning Apéry-like series
• Short walk adventures
• Genus-zero and genus-one string amplitudes and special multiple zeta values
• Sun's Series via Cyclotomic Multiple Zeta Values
• Explicit formulas of Euler sums via multiple zeta values
• New Evaluations of Inverse Binomial Series via Cyclotomic Multiple Zeta Values
• On a non-critical symmetric square $L$-value of the congruent number elliptic curves
• Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series